A Wigner-type Theorem on Symmetry Transformations in Banach Spaces

We obtain an analogue of Wigner’s classical theorem on symmetries for Banach spaces. The proof is based on a result from the theory of linear preservers. Moreover, we present two other Wigner-type results for finite dimensional linear spaces over general fields. Wigner’s theorem on symmetry transformations (sometimes called unitary-antiunitary theorem) plays fundamental role in quantum mechanics. This result can be formulated in several ways. For example, in [2, Theorem 3.1] the statement reads as follows. In the sequel P1(H) denotes the set of all rank-one (orthogonal) projections (or, in the language of quantum mechanics, the set of all pure states) on the Hilbert space H. We let tr stand for the usual trace-functional. Wigner’s theorem. Let H be a complex Hilbert space and let φ : P1(H) → P1(H) be a bijective function for which trφ(P )φ(Q) = trPQ (P,Q ∈ P1(H)). (1) Then there exists an either unitary or antiunitary operator U on H such that φ is of the form φ(P ) = UPU (P ∈ P1(H)). This formulation of Wigner’s theorem makes us possible to formulate an analogous theorem in the more general setting of Banach spaces, which we state below. Other Wigner-type theorems for Hilbert modules over matrix algebras or for indefinite inner product spaces or for type II factors can be found in our recent papers [6], [7], [8], respectively. If X is a Banach space, then X ′ denotes the (topological) dual of X. The set of all rank-one idempotents on X (which are the natural Banach space analogues of the projections) is denoted by I1(X). Now, our first result reads as follows. Date: February 8, 2008. 1991 Mathematics Subject Classification. Primary: 47N50.